The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability P∃no points in the Bessel process on(0,x1)∪(x2,x3)∪⋯∪(x2g,x2g+1),$$\begin{equation*} {\mathbb{P}\left(\exists \text{ no points in the Bessel process on}\ (0,{x}_{1})\cup ({x}_{2},{x}_{3})\cup \cdots \cup ({x}_{2g},{x}_{2g+1})\right),} \end{equation*}$$where 0


Introduction
Let g ≥ 0 be an integer, let x = (x 1 , x 2 , . . ., x 2g+1 ) be such that 0 < x 1 < • • • < x 2g+1 < +∞, and consider the Fredholm determinant where K| Ig is the trace class operator acting on L 2 (I g ) whose kernel is given by and J α is the Bessel function of the first kind of order α.
The determinantal point process on (0, +∞) associated to the kernel (1.2) is referred to as the Bessel point process.This is a universal point process in random matrix theory which models the behavior of the smallest eigenvalues for a wide class of large positive definite random matrices, see e.g.[37,39].As is well-known, F ( x) is the probability of finding no points on I g in the Bessel process.In this paper, we obtain asymptotics for F (r x) = F (rx 1 , . . ., rx 2g+1 ) as r → +∞.
Known results for g = 0. Tracy and Widom in [50] have shown that F (x 1 ) can be naturally expressed in terms of the solution of a Painlevé V equation.They also obtained large r asymptotics of F (rx 1 ), namely, as r → +∞, (1.3) where C is independent of r.Based on numerics, they conjectured that where G is Barnes' G-function.The expression (1.4) for the constant C was first established rigorously by Ehrhardt in [34] for α ∈ (−1, 1), and then by Deift, Krasovsky, and Vasilevska in [27] for all values of α ∈ (−1, +∞).
For g ≥ 1, it is known from [18] that F ( x) is related to a solution of a system of 2g + 1 coupled Painlevé V equations.However, to the best of our knowledge, there exist no results prior to this work on the large r asymptotics of F (r x) for g ≥ 1.
Statement of results.Let X be the Riemann surface of genus g associated to R(z), where (1.5) We view X as two copies of C that are glued along (−∞, −x 2g+1 ) ∪ • • • ∪ (−x 4 , −x 3 ) ∪ (−x 2 , −x 1 ).Given z ∈ C, we write z + ∈ X (resp.z − ∈ X) for the points on the first (resp.second) sheet of X that project onto z.We choose the sheets such that R(z + ) > 0 for z ∈ (−x 1 , +∞).Consider the cycles A 1 , . . ., A g , B 1 , . . ., B g shown in Figure 1.The cycle B j lies entirely in the first sheet and surrounds (−x 2j , −x 2j−1 ) in the positive direction, while A j lies partly in the upper half-plane of the first sheet (the solid red curves in Figure 1) and partly in the lower half-plane of the second sheet (the dashed red curves in Figure 1).These cycles form a canonical homology basis of X. Define the g × g matrix A and the column vector a by A = (a i,j ) g i,j=1 , a = (a 1,g+1 • • • a g,g+1 ) t , where a i,j = Ai s j−1 R(s) ds, i = 1, . . ., g, j = 1, . . ., g + 1, and t denotes the transpose operation.Because x 1 , . . ., x 2g+1 are real, it follows that a i,j ∈ R for all i = 1, . . ., g, j = 1, . . ., g + 1.Let ω = (ω 1 , . . ., ω g ) be the row vector of holomorphic one-forms given by and let q be the following polynomial of degree g: q j z j , where (q 0 , . . ., q g−1 ) t = − 1 2 A −1 a. (1.7)
Our first theorem provides an asymptotic formula for F (r x) for any choice of the points 0 where F is the Fredholm determinant (1.1), M > 0 is arbitrary but independent of r and C = C(M ) is independent of r.Furthermore, for any j = 1, . . ., 2g + 1, as r → +∞ we have Remark 1.If g = 0, then the term log θ( ν(r)) in (1.20) should be interpreted as being equal to 0.
where C is independent of r.
The case g = 1 is simpler because both (1.23) and (1.24) are automatically satisfied for any choice of 0 < x 1 < x 2 < x 3 < +∞.Hence, Corollary 1.4 always applies when g = 1, and we get the following result.(1.23).This follows from Liouville's theorem on diophantine approximation and the fact that √ 2 is a root of the degree 2 polynomial x 2 − 2.

Related work.
The determination of large gap asymptotics is a classical problem in random matrix theory with a long history.There exist various results on large gap asymptotics in the case of a gap on a single interval (the so-called "one-cut regime"), see [25,30,32,33,43,52] for the sine process, [2,24] for the Airy process, [27,34] for the Bessel process, [19,20,22] for the Wright's generalized Bessel and Meijer-G point processes, [21] for the Pearcey process, [3,[8][9][10][11][12][13][14][15][16][17]23] for thinned-deformations of these universal point processes, and [31,49,51] for the sine-β, Airy-β and Bessel-β point processes.We also refer to [44] and [38] for two overviews.Large gap asymptotics in the case of multiple intervals has been much less explored.For the sine process, this problem was first investigated by Widom [53], who obtained an explicit expression for the leading term in the asymptotics, and characterized the oscillations in terms of the solution to a Jacobi inversion problem.These oscillations were later described more explicitly by Deift, Its, and Zhou [26] in terms of θ-functions.Actually, the results of [26] are, in many respects, similar to the results of our first two theorems (Theorem 1.1 and Theorem 1.2).In particular, the asymptotic formula of [26] for the probability to observe g gaps in the sine process involves (1) an explicit leading term, (2) some oscillations of order 1 described in terms of θ-functions, and (3) a rather complicated integral involving ratios of θ-functions.As in our case (see (1.21)), the latter integral grows logarithmically with the size of the intervals.It was also noted in [26] that, in the generic case where a certain vector has "good Diophantine properties", this integral can be estimated with good control of the error term (this idea of [26] has been the main inspiration for our Theorem 1.2).However, the coefficient of the logarithmic term was written in [26] as a time average of ratios of θ-functions (cf.our formula (1.21)) and was not simplified.Recently, for the case of two intervals (this corresponds to a genus 1 situation), Fahs and Krasovsky in [35] showed that this coefficient is identically equal to −1/2.Large gap asymptotics for the Airy process in certain genus 1 situations were recently computed in [4,5,45], and there too simple expressions for the log coefficients appearing in the large gap asymptotics were obtained.We also mention that the multiplicative constants in the asymptotics were explicitly computed in [35,45].
In Theorem 1.3, we provide a simplified formula for the coefficient of the logarithmic term appearing in the large r asymptotics of F (r x) for an arbitrary g ≥ 0, in the generic case when the flow (1.22) is ergodic in R g /Z g .The proof of Theorem 1.3 requires the novel use of Birkhoff's ergodic theorem.More precisely, using the ergodicity of the flow (1.22), we rewrite the one-dimensional integrals B j of (1.21) (which we call time averages) as g-dimensional integrals over R g /Z g (which we call space averages).Quite remarkably, these space averages can in turn be evaluated explicitly after a non-trivial change of variables involving the Abel map, see Proposition 7.6 and Lemma 7.12.We expect that a similar simplification of the logarithmic term can be achieved by means of Birkhoff's ergodic theorem for any genus g ≥ 1 also for other point processes such as the sine and Airy processes.As noted above Corollary 1.5, the case g = 1 is somewhat simpler in several respects; in particular, for g = 1 every flow is both ergodic and has "good Diophantine properties".Hence there is no need to invoke Birkhoff's ergodic theorem when g = 1, see [4,5,35,45].For these reasons, we feel that Theorem 1.3 is an important novel contribution of the current paper.
Overview of the paper.In Section 2, we link the Fredholm determinant F (r x) to the solution Φ of a certain Riemann-Hilbert (RH) problem.In Sections 3-6, we perform an asymptotic analysis of Φ by means of the Deift/Zhou steepest descent method.This analysis is split into several sections as follows.In Section 3, we normalize the RH problem for Φ at infinity using an appropriate g-function, and then we proceed with the opening of the lenses.In Section 4, we construct the global parametrix P (∞) in terms of Riemann θ-functions.In Section 5, we construct local parametrices at z = −x j , j = 1, . . ., 2g + 1; these are built using the well-known solution of the Bessel model RH problem (see Appendix B).In Section 6, we complete our RH analysis with a small norm analysis.Section 7 relies on the results of Sections 3-6 and contains the proofs of Theorems 1.1-1.3.Birkhoff's ergodic theorem is used in the proof of Theorem 1.3.

Differential identity for F
We begin by establishing a relationship between the Fredholm determinant F (r x) and the solution to a particular Riemann-Hilbert problem.The goal of this section is to prove the following Proposition. where where where the principal branch is chosen for each fractional power.
(d) As z → −x j , j = 1, . . ., 2g + 1, we have e πiασ3 e πiασ3 where sgn is the signum function, G j is analytic in a neighborhood of −x j and satisfies det G j ≡ 1, V j (z) is defined by and H(z) is defined by ( (e) As z tends to 0, Φ takes the form where G 0 is analytic in a neighborhood of 0 and satisfies det G 0 ≡ 1.
Proof of Proposition 2.1.We follow the method developed by Its, Izergin, Korepin and Slavnov [41], and then further pursued in [26].Note from (1.2) that K can be rewritten as Also, from (1.1) and a direct rescaling, we get where K r | Ig is the trace-class integral operator acting on L 2 (I g ) whose kernel is given by Using well-known identities for trace-class integral operators, we get A direct computation using [40, eq. ( 8.401)] gives and it follows that the kernel of the integral operator where . Thus, by (2.8) we have On the other hand, where and so we conclude that It follows from [18, eq. ( 4.17)] that we can express Y in terms of Φ as follows where P Be , which is defined in [18, eq.(A.7)], satisfies as z → ∞, with and thus (2.11) The result now follows by using (2.10) and (2.11) in (2.9).
Our next goal is to obtain large r asymptotics of Φ 1,12 (r).For this, we will first perform a Deift-Zhou [29] steepest descent analysis on the RH problem for Φ.This analysis is carried out in Sections 3-6.

Steepest descent for Φ: first steps
In this section, we proceed with the first steps of the steepest descent method.We start by finding a g-function that will be used to normalize the RH problem for Φ at ∞.

g-function
The function R(z), defined in (1.5), satisfies where R(z) + and R(z) − denote the limits of R(s) as s → z from the upper and lower half-plane of the first sheet of X, respectively.We define the g-function by where the path of integration lies on the first sheet of X and does not cross (−∞, −x 1 ] + and coefficients q j , j = 0, . . ., g − 1 are given by (1.7).The following lemma extracts the necessary properties of the g-function.

The g-function satisfies the jump conditions
where Ω j > 0 is defined in (1.10).

As
where c is defined in (1.9).Proof.We begin with the observation that the second set of equations in (1.8) can be rewritten as where the path lies on the first sheet of X.Because R(z) is real and non-zero on (−x 2j+1 , −x 2j ) + , it follows from (3.6) that q has exactly one simple zero on each of the intervals (−x 2g+1 , −x 2g ), . .., 1.The fact that g is analytic in C \ (−∞, −x 1 ] follows directly from (3.2), and g(z) = g(z) follows from the fact that the coefficients of R and q are real.
4. It follows from the zeros of q(z) and (3.7) that Re g(z It follows from the definition (3.2) that g admits an expansion of the form g(z) for all |θ| ≤ π and j = 1, 2, . . .Since Re g(z) is harmonic on U s , its minimum value over the set U s is attained only on ∂U s , which by (3.9) and (3.10) is equal to 0. Thus Re g(z) > 0 for all z ∈ U s Since s ≥ ρ * was arbitrary, we conclude that Re g(z) > 0 for all z in the upper half-plane.The symmetry g(z) = g(z) now implies that Re g(z) > 0 for all z in the lower half-plane, which finishes the proof.

Rescaling of the RH problem
We now use the g-function (3.2) to define It can now be verified, using Lemma 3.1 and the RH problem for Φ, that T is the unique solution to the following RH problem.

RH problem for
is analytic, and we recall that Σ Φ is shown in Figure 2.
(b) The jumps for T are given by where we define where T 1 is independent of z and satisfies (e) As z → 0, T takes the form where T 0 is analytic in a neighborhood of 0.

Opening of the lenses
For each j ∈ {1, . . ., g}, let Σ j,+ and Σ j,− be two open curves starting at −x 2j , ending at −x 2j−1 , and lying in the upper and lower half-plane respectively, see also Figure 3.The bounded lens-shaped region delimited by Σ j,+ ∪ Σ j,− will be denoted by L j , j = 1, . . ., g.With the factorization (3.12) in mind, we define S in terms of T as follows It is straightforward to verify that S satisfies the following RH problem.

RH problem for
and Σ S is oriented as in Figure 3.

Global parametrix
In this section we construct the global parametrix P (∞) (z), which will serve as an approximation to S(z) when z is bounded away from the endpoints −x j , j = 1, . . ., 2g + 1.

RH problem for
(b) The jumps for P (∞) are given by (c) As z → ∞, we have where is independent of z.
As in [47], we proceed in two steps.First, we construct the solution P (∞) of an auxiliary RH problem which has no root-type singularity at the origin.Next, we consider a function D with a root-type behavior at 0 and suitable jumps.We will see that P (∞) is expressed in terms of P (∞) and D.
Consider the following RH problem: The jumps for P (∞) are given by (c) As z → ∞, we have where P (∞) is independent of z.
(d) As z → −x j , j = 1, . . ., 2g + 1, we have We express P (∞) in terms of Riemann θ-functions, whose definition and relevant properties are given in (1.15) and (1.16).Recall from (1.17) that the Abel map ϕ is given by and that ω is defined in (1.6).It will be convenient to also consider the following function, which we denote by ϕ C : where the path of integration lies on the first sheet.Recall that the period matrix τ is defined by (1.14).
It is well-known, see e.g.[36, p. 63], that τ is symmetric and has positive definite imaginary part.The following properties of ϕ C follow readily from its definition.
where e j denotes the jth column of the g × g identity matrix, e g+1 := 0, and τ j is the jth column of the matrix τ .Moreover, we have ϕ C (−x 1 ) = 0, ϕ C (∞) = 1 2 e 1 , and, for j = 1, . . ., g, Define where the branch is chosen such that β(z) > 0 for z > −x 1 , and define G : This periodicity property follows immediately from (1.16).Since −iτ is real and positive definite, θ( u) > 0 for all u ∈ R g , see [28,Section 4].In particular, G( u) > 0 for all u ∈ R g .Recall from (1.6) that the holomorphic differentials ω 1 , . . ., ω g are given by It can now be verified that where (A −1 ) g: denotes the g-th row of A −1 , ∇G is the gradient of G (of size g × 1), and ∂ j G denotes the j-th partial derivative of G.Moreover, Note that P (∞) and P (∞) have the same jumps on (−x 2j , −x 2j−1 ), j = 1, . . ., g + 1 but not on (−x 2j+1 , −x 2j ), j = 1, . . ., g.Let us introduce the following function where α0 := α 2 and the constants αj are defined by (1.12).The definition of the αj 's ensure that D(z) → 1 as z → ∞.It can be verified that D satisfies the following properties: where d 1 is given by (1.11).
We are now ready to solve the RH problem for P (∞) (z).
Proof.Using the properties of D(z), P (∞) (z), and the definition of ν 1 , . . ., ν g , we verify that the righthand side of (4.25) solves the RH problem for P (∞) .

. , g
Associated to the Jacobian variety J(X) are a total of 2 2g half-periods, 2 g−1 (2 g − 1) of which are odd half-periods, see e.g.[36, p. 303].It is well-known, see e.g.[36, p. 304], that the θ-function vanishes at each odd half-period.Proposition 4.1 allows us to express some of the odd half-periods in terms of the Abel map.
The radii of the disks are chosen sufficiently small, but independent of r, such that they do not intersect each other.Inside D −xj , P (−xj) has the same jumps as S, and is such that S(z)P (−xj) (z) −1 = O(1) as z → −x j .Furthermore, on the boundary of D −xj , we impose that P (−xj) "mathches" with P (∞) , in the sense that uniformly for z ∈ ∂D −xj .We will follow the pioneering work [26] and build these parametrices in terms of Bessel functions.We will show in Section 6 that P (−xj) (z) approximates S(z) for z ∈ D −xj .
(6.4) By the small-norm theory for RH problems [28,29], it follows that R exists for sufficiently large r and uniformly for z ∈ C \ Σ R .Using (6.3) and the relation we infer that It is easy to see from (6.4) that J ( R admits an analytic continuation from as z → −x j , j = 1, . . ., 2g + 1. Recalling that ∂D −xj is oriented in the clockwise direction, a simple residue calculation shows that, for z outside the disks, (6.7) can be rewritten as R ) and in particular, we have Let R 1 = lim z→∞ z(R(z) − I).From (6.9), we infer that The matrices (J −xj can be easily computed using (4.28), (4.31), (4.33), (5.3), (5.8), (6.4) and the fact that det P (∞) (z) ≡ 1.Thus, we have (J

R )
(−1) In particular, using (1.16), we find (J More precisely, the right-hand side of (6.11) for j = 1, . . ., g has been obtained after some simplifications using , and the right-hand side of (6.12) has been obtained using the relation G+ 2j = −e −2πi(ν1+•••+νj ) G− 2j .We note from (6.11) and (6.12) that the expressions for (J −xj 12 appear unrelated for odd and even j.In Proposition 7.8 below, we show that they can all (both for j even and odd) be rewritten in an unified way.

Proofs of Theorems 1.1-1.3
Recall from Proposition 2.1 that

Proof of Theorem 1.1
The proof relies on equation (7.1).We first compute the large r asymptotics of Φ 1,12 (r).
Our next goal is to rewrite 2i √ r as an r-derivative.To prepare ourselves for that matter, we first prove the following lemma.

Lemma 7.2. We have
Using the definition (1.10) of Ω j , and noting that a r,ℓ = Ar w ℓ , we get To prove (7.4), first note that the differential w g+1 is holomorphic except at ), and hence ).In terms of the local analytic coordinate u := z − 1 2 at z = ∞, this becomes as u → 0.
It follows that the differential η, defined by is an Abelian differential of the second kind with vanishing A-periods whose only pole is a double pole at z = ∞ (i.e. at u = 0); moreover, at this pole η has the singular part du/u 2 .Applying Riemann's bilinear identity to η and a holomorphic differential η, we find (see e.g.[36, p. 64, eq.(3.0.2)]) where f is a function such that df = η near u = 0. Let us fix j ∈ {1, . . ., g} and choose η = ω j .Then (7.6) reduces to where df j = ω j near u = 0. Using the definition (4.15) of ω j , we obtain The identity (7.4) follows by substituting (7.5) and (7.8) into (7.7).This completes the proof.
Proposition 7.3.We have the identity Proof.The first equality follows from (4.26).Note that G( 0) depends on r, whereas d 1 and (A −1 ) gj do not.Let z = (z 1 , . . ., z g ) ∈ C g .Using the definition (4.13) of G, we get Since θ( z) = θ(− z) for all z ∈ C g , we have ∂ j θ( 0) = 0 for all j = 1, . . ., g, and thus √ r , so we conclude that Finding a simplified expression for R 1,12 is more challenging and requires some preparation.A divisor on X is a formal product of points of the form where P 1 , . . ., P k ∈ X, α 1 , . . ., α k ∈ Z.
Let X g denote the n-fold symmetric product of the Riemann surface X, that is, X g is the set of all divisors of the form P 1 • • • P g where P 1 , . . ., P g ∈ X.It is known [36, section III.11.9] that X g can be given a complex structure, so that X g is a manifold of complex dimension g.The Abel map ϕ was defined in (1.17) as a map from X to J(X).Following [36, p. 92], we extend the map ϕ to X g as follows: Let π : X → C denote the projection of X onto C, and let A be the subset of X g consisting of all divisors D = P 1 • • • P g such that P j ∈ π −1 ([−x 2j+1 , −x 2j ]), j = 1, . . ., g.One can equip A with an atlas of smooth local charts inherited from the atlas of analytic local charts of X g , so that A is a smooth submanifold of X g of real dimension g.
We recall from [36, p. 110 (a)] that a divisor P 1 • • • P g is special if and only if there exists a non-zero holomorphic differential ω which vanishes at each of the points P 1 , . . ., P g .Lemma 7.4.Each divisor in A is non-special.
Proof.Since ω 1 , . . ., ω g form a basis of the space of holomorphic differentials, it follows that any holomorphic differential can be written in the form ω = p(z)dz/ R(z) where p is a polynomial of degree at most g − 1.If P 1 • • • P g ∈ A, then by definition of A we have P j1 = P j2 for 1 ≤ j 1 = j 2 ≤ g.Note that dz/ R(z) vanishes only at ∞. Hence, ω vanishes at P j if and only if p vanishes at P j .Since there exists no non-zero polynomial of degree ≤ g − 1 vanishing at g distinct points, the proof is complete.
Lemma 7.5.If e = ϕ(D) + K and D ∈ X g , then the multi-valued function X ∋ P → θ( ϕ(P ) − e) vanishes identically if and only if D is special.Furthermore, this function has a well-defined zero set, and if D is not special, then D is the divisor of zeros of X ∋ P → θ( ϕ(P ) − e).
Clearly, the set is a nonempty connected smooth submanifold of J(X) of real dimension g.Let ϕ| A denote the restriction of ϕ : X g → J(X) to A.
It readily follows from its definition that the map ϕ| A : A → T is smooth.We have shown in Lemma 7.4 that all divisors in A are non-special.Let D = P 1 • • • P g be an arbitrary divisor in A. Since D is non-special, the tangent map of ϕ : X g → J(X) at D has full rank (see e.g.[36, section VI.2.5]); hence the tangent map of ϕ| A : A → T at D also has full rank.Therefore, by the inverse function theorem, ϕ| A : A → T is a local diffeomorphism.
Since ϕ| A is continuous, A is compact, and T is Hausdorff, it follows that ϕ| A is a proper map [48,Proposition A.53].Therefore, we have shown that ϕ| A is a proper local diffeomorphism.Since A and T are nonempty and connected, this implies that ϕ| A is a smooth covering map [48,Proposition 4.46], that is, ϕ| A : A → T is a smooth surjective map, and for each t ∈ T , there exists a neighborhood U ⊆ T of t such that each component of ( ϕ| A ) −1 (U ) is mapped diffeomorphically onto U by ϕ| A .
Let µ ∈ R g /Z g .In view of Proposition 7.6, there exist D or equivalently 12) It follows from Lemma 7.4 that D and D are non-special, and from Lemma 7.5 that do not vanish identically and that their zero divisors are given by D and D, respectively.Let ι : X → X be the sheet-changing involution.Since ϕ(P ) = − ϕ(ι(P )) and K = −K mod L(X), it follows from (7.12) that Pk = ι( Pk ) (and in particular For each P ∈ X and µ ∈ R g /Z g , we have

Define the function
where P ∈ X is such that z = π(P ).The second equality in (7.15) follows from Proposition 7.7.
We are now ready to compute the large r asymptotics of . Proposition 7.9.As r → +∞, we have where M > 0 is independent of r.
We obtain (1.20) after substituting the asymptotics (7.16) into (7.1),integrating in r, and then exponentiating both sides.To finish the proof of Theorem 1.1, it remains to prove (1.21).Proposition 7.10.Let M > 0 and let H : R g /Z g → R be continuous.Then H ∈ R is well-defined by

.17)
In particular, ) In the same way that we use the analytic coordinate z to perform calculations on X (viewing X as a two-sheeted cover of the complex z-plane), we can use (z 1 , . . ., z g ) as coordinates on where z j denotes the coordinate on the j-th factor of X. Evaluating the integral on the right-hand side of (7.31) Using that ϕ ′ j (z)dz = ω j and that A k ω j = δ jk for j, k = 1, . . ., g, we conclude that .
By first adding the g-th column to the (g − 1)-th column, then the (g − 1)-th column to the (g − 2)-th column etc., we obtain = det(z 0 I − T ).
Since z 0 ∈ C was arbitrary, this finishes the proof.
Hence, we find + O(z −1 ) e 2z The unique solution of this RH problem exists and can be explicitly constructed in terms of Bessel functions [26,46].

Figure 2 :
Figure 2: The jump contour Σ Φ and the associated jump matrices in the case when g = 2.

Figure 4 :
Figure 4: The contour Σ R for R with g = 3.